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For each real $c$, some maximizer of $e(G)-c{k}_3(G)$ among graphs on $n$ vertices is complete multipartite graph. Among maximizers maximize ${\sum }_{v}d(v{)}^2$. Cloning either of two nonadjacent vertices preserves the objective, since maximality forces their local contributions $d(v)-ce(G[N(v)])$ to agree. If their vertex neighbourhoods differ, the sum of the two changes in the squared-degree of a vertex objective is twice the size of their symmetric difference, a contradiction. Nonadjacency therefore partitions the vertices into classes with identical vertex neighbourhoods.

Symmetrization for an arbitrary real coefficient. Among graphs maximizing $f(G)=e(G)-c{k}_3(G)$, choose one maximizingFor a vertex $x$, put $L(x)=d(x)-ce(G[N(x)])$. This is the contribution of edges and triangles in a graph containing $x$. If $u,v$ are nonadjacent, Zykov symmetrization replacing $u$ by a clone of $v$ changes $f$ by $L(v)-L(u)$; the opposite replacement changes it by $L(u)-L(v)$. Maximality implies $L(u)=L(v)$, so both changes preserve $f$.

Let $S=N(v)\setminus N(u)$ and $T=N(u)\setminus N(v)$. In the first replacement the vertex degrees in $S$ increase by one and those in $T$ decrease by one; in the opposite replacement these changes reverse. The changes to $d(u{)}^2+d(v{)}^2$ cancel when the two replacements are added. Since $(d+1{)}^2+(d-1{)}^2-2d^2=2$,If the two vertex neighbourhoods differ, one replacement increases $Q$, a contradiction. Thus every pair of nonadjacent vertices has the same vertex neighbourhood. Nonadjacency, with equality allowed, is an equivalence relation: if $u$ and $v$ are nonadjacent and $v$ and $w$ are nonadjacent, their identical vertex neighbourhoods forbid $uw$ as well. Its classes are independent sets and every cross-class edge is present. ThereforeThis edge-triangle symmetrization works for either sign of $c$.

The exact supporting line. For $n\ge 3$, writeWe prove the triangle support line between bipartite and tripartite Turan graphs by maximizing $e-c_{\ast }{k}_3$. First $c_{\ast }\ge 2/n$. For $n=3,4,5$, its values are respectively $1,1/2,1/2$. For $n\ge 6$, the exact balancing of the Turan graph gives $e_3\ge n^2/3-1/3$, while $e_2\le n^2/4$ and the arithmetic-geometric mean inequality gives $t_3\le n^3/27$. Hence

By edge-triangle symmetrization a maximizing graph can be taken complete multipartite graph; among such maximizers use one with the fewest nonempty parts. If there are at least four parts, let $a,b$ be the two smallest sizes. Then $a+b\le n/2$. Merging them loses $ab$ edges and $ab(n-a-b)$ triangles in a graph, so the objective changes byThis contradicts the choice of the number of parts. Thus at most three parts remain.

For three part sizes $a\ge b\ge d>0$, the objective isIf $1-c_{\ast }b\le 0$, merging the parts of sizes $a,d$ does not decrease it, again contradicting minimality. If $1-c_{\ast }b>0$ and $a-d\ge 2$, transferring one vertex from the largest to the smallest part raises $ad$ by $a-d-1>0$ and strictly increases the objective. Thus a three-part maximizer has all sizes differing by at most one, and is $T_3(n)$. A maximizer with at most two parts has objective at most $e_2$, since it has no triangles in a graph and the Mantel theorem bounds its edges. Finally,Therefore every graph satisfies $e(G)-c_{\ast }{k}_3(G)\le e_2$. At the prescribed edge count this givesFor $n\le 2$, the conclusion is simply the nonnegative lower bound zero. All rounding in the Turan graphs has been retained.

One edge above the bipartite threshold. Write $n=2m$. We prove the Triangle lower bound one edge above the Mantel threshold by induction on $m\ge 2$, allowing at least $m^2+1$ edges. For $m=2$, five edges on four vertices give two triangles in a graph, and adding edges preserves this bound.

We will also use the following consequence of an inductive bound: on $2a$ vertices, at least $a^2+q$ edges, for an integer $q\ge 1$, force at least $a+q-1$ triangles in a graph. Delete surplus edges first to leave exactly $a^2+q$. Repeatedly delete an edge in a triangle in a graph until $a^2+1$ edges remain. The Mantel theorem guarantees such a triangle in a graph at every step; each deletion destroys at least one distinct triangle in a graph. Apply the inductive bound to the remaining graph.

Now take exactly $m^2+1$ edges. If every edge lies in a triangle in a graph, the edge-triangle incidence bound gives $3k_3(G)\ge m^2+1$. This is at least $3m$ for $m\ge 3$; the base case was handled separately.

Otherwise choose an edge $uv$ in no triangle in a graph. Its endpoints have disjoint vertex neighbourhoods, so $d(u)+d(v)\le 2m$. Deleting $u,v$ removes exactly $d(u)+d(v)-1$ edges and leaves at least $(m-1{)}^2+1$ edges. If $d(u)+d(v)\le 2m-1$, it leaves at least $(m-1{)}^2+2$ edges, and the strengthened inductive bound gives at least $m$ triangles in a graph already.

If $d(u)+d(v)=2m$, induction gives at least $m-1$ triangles in a graph after deletion. There must be an additional triangle in a graph containing $u$ or $v$. Otherwise both $N(u)$ and $N(v)$ are independent sets; since they are disjoint and cover all vertices, $G$ would be bipartite graph, contradicting $e(G)>m^2$ by the Mantel theorem. HenceThe bound is sharp: add one internal edge to one class of the complete bipartite graph $K_{m,m}$. Its triangles in a graph are precisely that edge together with each of the $m$ vertices in the other class.

Turan theorem states that among all $N$-vertex graphs containing no $K_{r+1}$, the maximum number of edges is attained by the Turan graph $T_r(N)$, the complete multipartite graph whose $r$ part sizes differ by at most one.

Here is the Zykov symmetrization proof. Start with an extremal $K_{r+1}$-free graph. If nonadjacent vertices $u,v$ have different neighbourhoods, replace the lower-degree vertex by a clone of the higher-degree one: delete all its incident edges and join it to precisely the neighbours of the other vertex. This does not decrease the number of edges and cannot create a $K_{r+1}$, because a clique using the clone corresponds to one using the original vertex. Repeating the operation, with the standard tie-breaking that merges equal-neighbourhood classes, produces a complete multipartite extremal graph. It has at most $r$ nonempty parts, since choosing one vertex from each part gives a clique.

If two part sizes satisfy $a\ge b+2$, moving one vertex from the larger part to the smaller changes the number of cross-edges byThus an extremal partition has exactly $r$ parts whose sizes differ by at most one. This is $T_r(N)$ and proves the theorem. For $r=2$ it is Mantel theorem: a triangle-free graph on $N$ vertices has at most $\lfloor N^2/4\rfloor$ edges.

The Turan graph $T_r(n)$ is the complete multipartite graph on $n$ vertices with $r$ parts whose sizes differ by at most one.